Download Composite Optical Vortices Formed by Collinear Laguerre-Gauss Beams

Composite Optical Vortices Formed by Collinear
Laguerre-Gauss Beams
E.J. Galvez, N. Smiley, and N. Fernandes
Department of Physics and Astronomy, Colgate University, 13 Oak Drive, Hamilton, New York
13346, U.S.A.
ABSTRACT
We study the optical fields that are produced when two Laguerre Gauss beams, each carrying an optical vortex,
are superimposed collinearly. We find that the resulting beam contains new vortices. The number of vortices
and their location depends on the charge of the vortices of the component beams and the relative intensity of
the two beams. Our presentation focuses on the cases where the component Laguerre-Gauss beams are of order
one and two.
Keywords: Singular optics, Optical Vortices, Laguerre-Gauss Beams, Orbital Angular Momentum
1. INTRODUCTION
The study of optical beams bearing phase singularities is interesting from fundamental and applied perspectives.
Rich in physical phenomena, optical singularities provide an important setting for discovering new wave phenomena. The angular momentum carried by beams with phase vortices also adds a new dimension to the topic.
Phase vortices have their main application in the transfer of orbital angular momentum from light to matter
in optical tweezers. More recently, it has been shown that the quantization of this angular momentum has the
potential for encoding multiple quantum bits of information.
Recent research on optical vortices has revealed novel results on the propagation of vortex structures, with
findings such as the knotting of vortex filaments. The optical wave field in these studies was created by
the superposition of beams that contain fundamental phase singularities, such as Laguerre-Gauss (LG) beams.
Thus, the study of the superposition of beams bearing fundamental vortices is an important step for a better
understanding of the problem. A recent numerical study of the superposition of LG beams of the same order
showed an interesting redistribution of the phase vortices as a function of the lateral displacement of the two
beams. Here we study both theoretically and experimentally the optical fields that result when we combine two
LG beams of different order. In contrast to the previous work, we keep the two component LG beams collinear,
but vary other parameters, such as the order of the components and their relative intensity. We examine the
two cases where one component LG beam is of order one and the other beam is of order two.
This article is organized as follows. In Sec. 2 we present analytical and computational analyses of the problem.
Section 3 describes the experimental technique that we used to produce and diagnose the composite wave fields,
and in Sec. 4 we present the imaging of the composite beams.
1
2
3
4, 5
6
2. THEORY
Let µp be the field of an LG beam of order N = 2p + | |, where p and are the radial and azimuthal indices,
respectively. If we consider the case where p = 0, then the field distribution of the beam is a single ring that
contains a phase vortex of charge at its center. The normalized field in this beam is given by7
√
2 )1 /2 1 ( r 2 )
µ0 = (
π | |!
w w
| |
e−r
2 /w 2
e
−i φ ei[kz−kr2 /(2R)] eiϕ ,
Further author information: (Send correspondence to E.J.G.)
E.J.G.: E-mail: [email protected], Telephone: 1 315 228 7205
(1)
Figure 1. Simulations of a composite optical vortex beam made of component beams with 1 = +1 and 2 = +2. Row
(a) shows the distribution of intensity on a gray scale, going from low (black) to high (white) intensity. Row (b) shows
the phase of the field; the gray scale varies from white representing a phase of zero to black representing 2π. The angle
theta specifies the relative weight of the two component vortices, as determined by Eq. 2.
where r, φ and z are the cylindrical coordinates, w is the beam radius, k is the wave number, R is the radius of
curvature of the wave-front, ϕ = (N + 1) tan−1 (z/zR ) is the Gouy phase, and zR is the Raleigh range.
Now consider the collinear superposition of two LG beams with charge 1 and 2 . Let us set |2 | > |1 |. The
total field is given by
(2)
µT = sin θ µ01 + cos θ µ02 eiδ ,
where δ is the relative phase between the two beams and θ is a parameter that specifies the relative weight of
the two beams. The location of the phase vortices in this field is obtained by finding the points (r, φ) where µT
meets the conditions:8
Re(µT ) = 0
Im(µT ) = 0.
(3)
(4)
We find that the solution yields a vortex of charge 1 located at r = 0, and |2 − 1 | vortices of charge 2 /|2 |
located at positions specified by the coordinates (r, φ), where
r
=√
w
2
/
1 (| 2 |−| 1 |)
| 2 |!
| 1 |!
and
φ=
tan θ
δ + nπ
,
2 1
−
,
(5)
(6)
with n being an odd integer.
It is interesting to understand how this works. The variable θ defines the weight of each component beam.
When θ = 0 it is purely an 2 -vortex beam, and when θ = π/2 it is purely an 1 -vortex beam. That is, at
these two extremes the beam is composed of a vortex located at the center of the beam, per definition of the
Laguerre-Gauss beams in Eq. 1. For values of θ between 0 and π/2 the beam profile has a much more complex
structure. It is consistent with the following view: at θ = 0 we have a central 2 vortex that is composed of the 1
vortex superimposed with | 2 − 1 | singly charged vortices with a combined charge 2 − 1 . As θ is increased from
zero, the | 2 − 1 | vortices move away from the center of the beam reaching their limiting positions at θ = π/2,
where r = ∞. Thus, at this limit there is a single vortex of charge 1 at the center of the beam.
We have verified the solution presented above with numerical calculations of the total field in a transverse
plane of the beam. For simplicity these calculations assumed z = 0, which imply R = ∞ and ϕ = 0. The
calculated two-dimensional images, shown in Figs. 1 and 2, also allow a better appreciation of the structure of
the composite beams. Each simulation is displayed in a two-dimensional frame of length 4w.
Simulations of a composite optical vortex beam made of component beams with 1 = +1 and 2 = −2. Row
(a) shows the distribution of intensity on a gray scale, going from low (black) to high (white) intensity. Row (b) shows
the phase of the field; the gray scale varies from white representing a phase of zero to black representing 2π. The angle
theta specifies the relative weight of the two component vortices, as determined by Eq. 2.
Figure 2.
Let us start with the simplest situation that we have found: 1 = +1 and 2 = +2. The calculations can be
seen in Fig. 1, which displays in row (a) the intensity profile of the beam, and in row (b) the phase of the field.
In both types of images we have deliberately discretized the gray scale to appreciate better the intensity and
phase contours. One step in the gray scale corresponds to one tenth of the full scale. Thus, in the case of the
phase plots each step of the gray scale corresponds to a phase increment of π/5. The convention for the sign of
the vortex is the following: a vortex charge is positive when the shading increases in darkness when going in a
counter clockwise manner around the vortex (e.g., the phase plots for θ = 90◦ in Figs. 1 and 2 contain a single
+1 vortex).
The analytical solution for this case specifies that there are two +1 vortices. At θ = 0 they are both at
r = 0. As θ increases from zero we see one +1 vortex moving linearly away from the center of the beam. It is
interesting to see that as θ increases the intensity pattern opens up giving the appearance of expelling the +1
vortex, and closing back up once the leaving vortex is away from the center. Notice that the intensity pattern is
very sensitive to the location of the leaving vortex: at θ = 67.5◦ this vortex is at r = 2.4w, and thus is outside of
the plot frame, and in a location where the intensity of the beam is negligibly small. However, despite this the
intensity distribution of the beam remains highly asymmetric. When θ = 90◦ the beam is left with a +1 vortex
at its center.
In the computations of Figs. 1 and 2 we set the relative phase δ between the two beams to zero. The
orientation of the pattern depends on δ: as δ is varied the entire pattern rotates about the center of the beam.
We used this feature in the laboratory to align the component beams collinear.
The case where 1 = +1 and 2 = −2 is much more interesting. The solution specifies a +1 vortex at r = 0
with three −1 vortices moving radially outwards as θ is increased from zero. When θ = 22.5◦ and δ = 0, Eqs. 5
and 6 predict three −1 vortices located at the radial distance r = 0.41w and symmetrically oriented at the angles
φ = π/3, φ = π and φ = 5π/3. This analytical solution is verified by the calculations of Fig. 2. The retreating
vortices are more obvious when θ = 45◦ . As shown in row (a) of Fig. 2, the intensity is zero at the locations of
the vortices.
Let us analyze the phase variation in Fig. 2. Although the pattern should be symmetric under a rotation of
2π/3, the discretization of the gray scale introduces some minor artificial asymmetries. It is interesting to note
that the phase variation around the vortices is not uniform. The phase gradient is highest along the lines that
connect the vortices. It is also interesting that in the regions of non-zero intensity the phase is not constant, and
the gradient of the phase depends on the value of θ.
We have also investigated other possibilities. Similar behavior is observed for other values of 1 and 2 . When
|2 − 1 | = 1 one singly charged vortex moves radially away from the center of the beam for increasing values
of θ, leaving a vortex of charge 1 at the center. This behavior is similar to the first case that we discussed.
Figure 3. Optical layout of the apparatus to study composite optical vortices. Beams with 2 = −2 and = +1 vortices
are produced by forked diffraction gratings and superposed in a Mach-Zehnder interferometer. Individual components
shown explicitly are half-wave plates (H), polarizers (P), beam blockers (B), lens (L) and neutral-density filters (F).
When
|
−
| > 1 the composite vortices are arranged symmetrically about the center of the beam, and moving
radially away from it as θ is increased. This is similar to the second case discussed above.
When the two beams have the same order (i.e., = − ) no new vortices appear. In these cases only
the center of the beam has a vortex, if any. Then as θ is varied only the phase readjusts. For θ < π/4 the
phase increases nonlinearly and the intensity distributes into 2| | lobes. The phase increases in the sense of
the dominant singularity , with its gradient being highest in the regions between the lobes. When θ = π/4
the regions between the lobes become radial nodes, and the phase jumps discontinuously by π at the node lines.
Thus, at this value of θ the central phase singularity disappears and the nodes contain shear phase singularities.
For θ > π/4 the continuous phase gradients reappear but in the opposite sense, signaling the prevalence of the
oppositely-charged phase vortex (i.e., ). A common example of this is the case where = +2 and = −2.
When θ = π/4 they give rise to a mode that has four lobes, which turns out to be the well known Hermite-Gauss
mode “11” (i.e., TEM ).
2
1
1
2
2
2
1
11
2
1
7, 9
RIMENTAL ARRANGEMENT
3. EXPE
The optical setup used to perform the experiments consisted of several nested interferometers. Figure 3 shows a
schematic of the optical layout. A light beam from a HeNe laser was split by 50-50 non-polarizing beam splitter
cubes into four beams. After preparation into the desired LG modes, the beams were recombined coherently
downstream via a second set of non-polarizing beam splitters. A charge-coupled-device (CCD) camera imaged
the final beam.
The incident beam was prepared to be linearly polarized along a vertical plane. The first beam that split off
the main laser beam was used as a diagnosis tool in the preparation of the individual LG beams. In a second
role, it was expanded with a diverging lens and deliberately misaligned. The resulting beam was used to create
a fringe interference pattern with the composite beam. This served to identify the optical vortices, as described
below.
The second beam that split off the main beam from the laser was sent to a computer-generated charge-1
forked binary amplitude grating. A steering mirror and an iris placed after the grating were used to select the
= +1 beam. A half-wave plate (H in Fig. 3) placed in the path of the beam rotated its polarization by 90
degrees.
The light that remained in the main beam axis continued to a charge-2 forked diffraction grating. A MachZehnder interferometer placed after it was used to channel the first-order, = +2 and = −2, LG beams to each
Figure 4. Images of the composite beams for the case where 1 = +1 and 2 = +2. Row (a) shows images of the
composite beam, row (b) shows the interference of the composite beam with a tilted and expanded reference beam, and
row (c) shows the phase computations with the phase between the component beams δ adjusted to match the experimental
image.
of its arms. For the experiments reported here we selected one of the | | = 2 beams by putting a beam block (B
in Fig. 3) in the appropriate arm of the interferometer. The pure LG beams were diagnosed by interference with
the = 0 reference beam; the gratings were mounted on XY translation stages so they could be centered on the
beam without affecting the beam alignment.
The vertically polarized | | = 2 beam was combined with the horizontally polarized = +1 beam with a
non-polarizing beam splitter. The intensities of each beam were adjusted to be equal by means of neutral density
filters (F in Fig. 3). The two beams with polarizations orthogonal to each other were made interfere by projecting
them along the same polarization plane with a linear polarizer. The polarizer (P in Fig. 3) had its transmission
axis oriented an angle θ relative to the vertical. It is interesting that because of the axial phase of both beams,
the intensity of the resulting beam remained the same regardless of the value of θ. This served as a useful tool
for adjusting the relative intensity of the beams while keeping the total intensity constant.
The resulting beam, called hereafter the signal beam, was attenuated by neutral density filters and imaged
by the CCD camera. A critical aspect of the alignment was the combination of the two component beams. We
used the inherent symmetry of the patterns as an aid in their alignment. One of the mirrors steering the 1
beam was mounted on a translation stage that had a piezo-electric ceramic as a spacer. As a voltage applied to
the piezo-electric was increased, the position of the mirror mounted on the stage was changed slightly, changing
the phase δ between the two component beams. This feature allowed the observation of the signal beam as a
function of δ. If the beams were aligned perfectly collinear then the resulting pattern rotated about the axis of
the beam as δ was changed.
To locate the position of the composite optical vortices we imaged the fringe pattern formed by the interference
of the signal beam with the reference beam. For maximum contrast, the polarization of the signal beam was
rotated to be in the same (vertical) plane as the polarization of the reference beam. We did this rotation by
means of a half-wave plate. Phase vortices appeared as forks in the images, as will be shown in the next section.
4. RESULTS
Figure 4 shows a sequence of images for the case where the vortices of the component LG beams were 2 = +2
and 1 = +1. Row (a) shows images of the composite beams for different settings of θ. Row (b) shows the
Images of the composite beams for the case where 1 = +1 and 2 = −2. Row (a) shows images of the
composite beam, row (b) shows the interference of the composite beam with a tilted and expanded reference beam and
row (c) shows the phase computations with the phase between the component beams δ adjusted to match the experimental
image.
Figure 5.
interference patterns of the reference beam with the corresponding composite beam. In row (c) we display the
phase plots for the corresponding cases; this time with a gray scale of 100 steps. The reader must not get
distracted by the sharp white/black boundaries. They represent a step of π/50. One can identify the vortices
by the forks that appear in the interference pattern. The sign of the vortex is specified by the orientation of the
forks in the figure: a positive charge appears as a fork with the fork tines pointing toward the lower right.
When θ = 0 we see two +1 forks in the central area of the image. We do not observe the expected single
+2-charge fork. Although we have not investigated this in great detail, we speculate that it might be due to slight
deviations of the component beams from the ideal LG form caused by the transmission of the beams through
imperfect optical elements.
The images in Fig. 4 show qualitatively the expected changes in the positions of the vortices when θ was
increased. A similar confirmation is observed in Fig. 5, for the case where 1 = +1 and 2 = −2. The values of
θ chosen for both figures correspond to images that exhibit appreciable changes in the locations of the vortices.
Note the correlation between the orientation of the forks and the expected sign of the vortices. Because of the
coarse size of the forks we were not able to make a stringent quantitative comparison of the vortex locations
with the theoretical predictions.
Figure 6 shows a blown-up image of the interference pattern for the case where 1 = +1 and 2 = −2. Three
forks of charge −1 appear at the ends of a tilted “Y”, with a fork of charge +1 located at its center. The lack
of perfect symmetry is due to a slight misalignment of the component beams. The figure also allows one to
appreciate the limitations of this method of imaging the vortices: the uncertainty in the position of the forks is
a significant fraction of their radial position.
5. SUMMARY AND CONCLUSIONS
In summary, we have studied both experimentally and theoretically the composite vortices that arise when two
high-order LG beams bearing optical vortices are combined collinearly. For component beams of different order
we observe the creation of new vortices, which appear at locations determined by the value of the charge of
the component beams and their relative intensity. We find the arrangement of composite vortices to follow a
universal pattern. At the center of the beam there is a vortex of charge equal to the one of the component beam
Images of an interference pattern between the composite beam for the case
and the = 0 reference beam.
Figure 6.
1
= +1 and 2 = −2 when θ = 45◦ ,
that has the lowest charge magnitude, surrounded by one or more singly-charged vortices. The total charge of all
the vortices, including the central one, adds to the charge of the component beam that has the greatest charge
magnitude.
When the composite beams are of the same order our calculations show that no new vortices appear. Rather,
the phase distributes unevenly around the central vortex. When the weights of the two beams are equal the
central singularity disappears, giving rise to shear phase singularities.
This work was limited to the case of two collinear component beams. It will be interesting to extend this
work to the systematic study of the non-collinear case. In such a case we expect to find the composite vortices
in non symmetric arrangements. Other possibilities for further work include the combination of more that two
beams, where the created vortices may not be singly charged. Although the number of parameters is larger than
the case studied here, the resulting patterns may be richer.
ACKNOWLEDGMENTS
This work was supported in part by a Schlichting Fellowship of Colgate University.
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